Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved.[1] These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.
This article is a composite of unsolved problems derived from many sources, including but not limited to lists considered authoritative. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.
Lists of unsolved problems in mathematics
Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.
List Number of problems Number unresolved
or incompletely resolved Proposed by Proposed in
Hilbert's problems[2] 23 15 David Hilbert 1900
Landau's problems[3] 4 4 Edmund Landau 1912
Taniyama's problems[4] 36 - Yutaka Taniyama 1955
Thurston's 24 questions[5][6] 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize problems 7 6[7] Clay Mathematics Institute 2000
Simon problems 15 <12[8][9] Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[10] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[11][12] 23 - DARPA 2007
Millennium Prize Problems
Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of July, 2020:[7]
P versus NP
Hodge conjecture
Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture
The seventh problem, the Poincaré conjecture, has been solved;[13] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[14]
Unsolved problems
Algebra
In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.
Homological conjectures in commutative algebra
Finite lattice representation problem
Hilbert's sixteenth problem
Hilbert's fifteenth problem
Hadamard conjecture
Jacobson's conjecture
Crouzeix's conjecture
Existence of perfect cuboids and associated cuboid conjectures
Zauner's conjecture: existence of SIC-POVMs in all dimensions
Wild problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
Köthe conjecture
Birch–Tate conjecture
Serre's conjecture II
Bombieri–Lang conjecture
Farrell–Jones conjecture
Bost conjecture
Rota's basis conjecture
Uniformity conjecture
Kaplansky's conjectures
Kummer–Vandiver conjecture
Serre's multiplicity conjectures
Pierce–Birkhoff conjecture
Eilenberg–Ganea conjecture
Green's conjecture
Grothendieck–Katz p-curvature conjecture
Sendov's conjecture
Zariski–Lipman conjecture
The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.[15]
The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.[16]
Analysis
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.
The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals[17]
Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials[18]
The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy[19]
Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals[17]
Are γ {\displaystyle \gamma } \gamma (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan's constant, or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[20][21][22]
Vitushkin's conjecture
Invariant subspace problem
Kung–Traub conjecture[23]
Regularity of solutions of Vlasov–Maxwell equations
Regularity of solutions of Euler equations
Convergence of Flint Hills series
Combinatorics
Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets[24]
The lonely runner conjecture: if k + 1 {\displaystyle k+1} k+1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1 / ( k + 1 ) {\displaystyle 1/(k+1)} 1/(k+1) from each other runner) at some time?[25]
Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[26]
Finding a function to model n-step self-avoiding walks.[27]
The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?[28]
Give a combinatorial interpretation of the Kronecker coefficients.[29]
Open questions concerning Latin squares
The values of the Dedekind numbers M ( n ) {\displaystyle M(n)} M(n) for n ≥ 9 {\displaystyle n\geq 9} {\displaystyle n\geq 9}.[30]
The values of the Ramsey numbers, particularly R ( 5 , 5 ) {\displaystyle R(5,5)} R(5,5)
The values of the Van der Waerden numbers
Dynamical systems
A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.
Collatz conjecture (3n + 1 conjecture)
Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?
Furstenberg conjecture – Is every invariant and ergodic measure for the × 2 , × 3 {\displaystyle \times 2,\times 3} \times 2,\times 3 action on the circle either Lebesgue or atomic?
Margulis conjecture – Measure classification for diagonalizable actions in higher-rank groups
MLC conjecture – Is the Mandelbrot set locally connected?
Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?
Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
Eremenko's conjecture that every component of the escaping set of an entire transcendental function is unbounded
Is every reversible cellular automaton in three or more dimensions locally reversible?[31]
Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?[32]
Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
Quantum unique ergodicity conjecture[33]
Berry–Tabor conjecture
Painlevé conjecture
Games and puzzles
Combinatorial games
Sudoku:
What is the maximum number of givens for a minimal puzzle?[34]
How many puzzles have exactly one solution?[34]
How many puzzles with exactly one solution are minimal?[34]
Tic-tac-toe variants:
Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?[35]
What is the Turing completeness status of all unique elementary cellular automata?
Games with imperfect information
Rendezvous problem
Geometry
Algebraic geometry
Abundance conjecture
Bass conjecture
Deligne conjecture
Dixmier conjecture
Fröberg conjecture
Fujita conjecture
Hartshorne conjectures[36]
The Jacobian conjecture
Manin conjecture
Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory[37]
Nakai conjecture
Resolution of singularities in characteristic p {\displaystyle p} p
Standard conjectures on algebraic cycles
Section conjecture
Tate conjecture
Termination of flips
Virasoro conjecture
Weight-monodromy conjecture
Zariski multiplicity conjecture[38]
Differential geometry
The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[39]
The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[40]
The spherical Bernstein's problem, a possible generalization of the original Bernstein's problem
Cartan–Hadamard conjecture: Can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds?
Carathéodory conjecture
Chern's conjecture (affine geometry)
Chern's conjecture for hypersurfaces in spheres
Yau's conjecture
Yau's conjecture on the first eigenvalue
Closed curve problem: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[41]
Discrete geometry
In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.
Solving the happy ending problem for arbitrary n {\displaystyle n} n[42]
Finding matching upper and lower bounds for k-sets and halving lines[43]
The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[44]
Find lower and upper bounds for Borsuk's problem on the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
The Kobon triangle problem on triangles in line arrangements[45]
The McMullen problem on projectively transforming sets of points into convex position[46]
Tripod packing[47]
Ulam's packing conjecture about the identity of the worst-packing convex solid[48]
Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
What is the asymptotic growth rate of wasted space for packing unit squares into a half-integer square?[49]
Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[50]
How many unit distances can be determined by a set of n points in the Euclidean plane?[51]
Opaque forest problem
Improving lower and upper bounds for the Heilbronn triangle problem.
Euclidean geometry
Bellman's lost in a forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation[52]
Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?[53]
Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?[54]
Dissection into orthoschemes – is it possible for simplices of every dimension?[55]
The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?[56]
The Erdős–Oler conjecture that when n {\displaystyle n} n is a triangular number, packing n − 1 {\displaystyle n-1} n-1 circles in an equilateral triangle requires a triangle of the same size as packing n {\displaystyle n} n circles[57]
Falconer's conjecture that sets of Hausdorff dimension greater than d / 2 {\displaystyle d/2} d/2 in R d {\displaystyle \mathbb {R} ^{d}} \mathbb {R} ^{d} must have a distance set of nonzero Lebesgue measure[58]
Inscribed square problem, also known as Toeplitz' conjecture – does every Jordan curve have an inscribed square?[59]
The Kakeya conjecture – do n {\displaystyle n} n-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to n {\displaystyle n} n?[60]
The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem[61]
Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one[62]
Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?[63]
The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[64]
Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?[65][66]
The Thomson problem – what is the minimum energy configuration of n {\displaystyle n} n mutually-repelling particles on a unit sphere?[67]
Uniform 5-polytopes – find and classify the complete set of these shapes[68]
Covering problem of Rado – if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[69]
Atiyah conjecture on configurations
Graph theory
Paths and cycles in graphs
Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[70]
Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[71]
The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[72]
The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[73]
The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[74]
The Lovász conjecture on Hamiltonian paths in symmetric graphs[75]
The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[76]
Graph coloring and labeling
An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.
Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs[77]
The Erdős–Faber–Lovász conjecture on coloring unions of cliques[78]
The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree[79]
The Hadwiger conjecture relating coloring to clique minors[80]
The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[81]
Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[82]
The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[83]
The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[84]
Graph drawing
The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[85]
The Blankenship–Oporowski conjecture on the book thickness of subdivisions[86]
Conway's thrackle conjecture[87]
Harborth's conjecture that every planar graph can be drawn with integer edge lengths[88]
Negami's conjecture on projective-plane embeddings of graphs with planar covers[89]
The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[90]
Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[91]
Universal point sets of subquadratic size for planar graphs[92]
Word-representation of graphs
Characterise (non-)word-representable planar graphs [93][94][95][96]
Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [97])
Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter [98]
Is the line graph of a non-word-representable graph always non-word-representable? [93][94][95][96]
Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter? [93][94][95][96]
Is it true that out of all bipartite graphs crown graphs require longest word-representants? [99]
Characterise word-representable graphs in terms of (induced) forbidden subgraphs. [93][94][95][96]
Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)? [93][94][95][96]
Miscellaneous graph theory
Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?[100]
The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[101]
The GNRS conjecture on whether minor-closed graph families have ℓ 1 {\displaystyle \ell _{1}} \ell _{1} embeddings with bounded distortion[102]
Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs[103]
The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[104]
Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[105]
Meyniel's conjecture that cop number is O ( n ) {\displaystyle O({\sqrt {n}})} O({\sqrt n})[106]
Does a Moore graph with girth 5 and degree 57 exist?[107]
What is the largest possible pathwidth of an n-vertex cubic graph?[108]
The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.[109][110]
The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[111]
Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?[112]
Sumner's conjecture: does every ( 2 n − 2 ) {\displaystyle (2n-2)} (2n-2)-vertex tournament contain as a subgraph every n {\displaystyle n} n-vertex oriented tree?[113]
Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow[114]
Vizing's conjecture on the domination number of cartesian products of graphs[115]
Group theory
The free Burnside group B ( 2 , 3 ) {\displaystyle B(2,3)} {\displaystyle B(2,3)} is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups B ( m , n ) {\displaystyle B(m,n)} {\displaystyle B(m,n)} are finite remains open.
Is every finitely presented periodic group finite?
The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Is every group surjunctive?
Andrews–Curtis conjecture
Herzog–Schönheim conjecture
Does generalized moonshine exist?
Are there an infinite number of Leinster groups?
Guralnick–Thompson conjecture[116]
Problems in loop theory and quasigroup theory consider generalizations of groups
The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[117]
Model theory and formal languages
Vaught's conjecture
The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ℵ 0 {\displaystyle \aleph _{0}} \aleph _{0} is a simple algebraic group over an algebraically closed field.
The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ℵ 1 {\displaystyle \aleph _{1}} \aleph _{1}-saturated models of a countable theory.[118]
Determine the structure of Keisler's order[119][120]
The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
Is the theory of the field of Laurent series over Z p {\displaystyle \mathbb {Z} _{p}} \mathbb {Z} _{p} decidable? of the field of polynomials over C {\displaystyle \mathbb {C} } \mathbb {C} ?
(BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[121]
The Stable Forking Conjecture for simple theories[122]
For which number fields does Hilbert's tenth problem hold?
Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ℵ ω 1 {\displaystyle \aleph _{\omega _{1}}} \aleph _{\omega _{1}} does it have a model of cardinality continuum?[123]
Shelah's eventual categoricity conjecture: For every cardinal λ {\displaystyle \lambda } \lambda there exists a cardinal μ ( λ ) {\displaystyle \mu (\lambda )} {\displaystyle \mu (\lambda )} such that If an AEC K with LS(K)<= λ {\displaystyle \lambda } \lambda is categorical in a cardinal above μ ( λ ) {\displaystyle \mu (\lambda )} {\displaystyle \mu (\lambda )} then it is categorical in all cardinals above μ ( λ ) {\displaystyle \mu (\lambda )} {\displaystyle \mu (\lambda )}.[118][124]
Shelah's categoricity conjecture for L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} L_{\omega_1,\omega}: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[118]
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[125]
If the class of atomic models of a complete first order theory is categorical in the ℵ n {\displaystyle \aleph _{n}} \aleph _{n}, is it categorical in every cardinal?[126][127]
Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
Kueker's conjecture[128]
Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
Do the Henson graphs have the finite model property?
The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[129]
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[130]
Generalized star height problem
Number theory
General
6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.
Grand Riemann hypothesis
Generalized Riemann hypothesis
Riemann hypothesis
n conjecture
abc conjecture
Szpiro's conjecture
Hilbert's ninth problem
Hilbert's eleventh problem
Hilbert's twelfth problem
Carmichael's totient function conjecture
Erdős–Straus conjecture
Erdős–Ulam problem
Pillai's conjecture
Hall's conjecture
Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)
Montgomery's pair correlation conjecture
Hilbert–Pólya conjecture
Grimm's conjecture
Leopoldt's conjecture
Scholz conjecture
Do any odd perfect numbers exist?
Are there infinitely many perfect numbers?
Do quasiperfect numbers exist?
Do any odd weird numbers exist?
Do any Lychrel numbers exist?
Is 10 a solitary number?
Catalan–Dickson conjecture on aliquot sequences
Do any Taxicab(5, 2, n) exist for n > 1?
Brocard's problem: existence of integers, (n,m), such that n! + 1 = m2 other than n = 4, 5, 7
Beilinson conjecture
Littlewood conjecture
Vojta's conjecture
Goormaghtigh conjecture
Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
Are there infinitely many amicable numbers?
Are there any pairs of amicable numbers which have opposite parity?
Are there any pairs of relatively prime amicable numbers?
Are there infinitely many betrothed numbers?
Are there any pairs of betrothed numbers which have same parity?
The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
Piltz divisor problem, especially Dirichlet's divisor problem
Exponent pair conjecture
Is π a normal number (its digits are "random")?[131]
Casas-Alvero conjecture
Sato–Tate conjecture
Find value of De Bruijn–Newman constant
Which integers can be written as the sum of three perfect cubes?[132]
Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?
Is there a covering system with odd distinct moduli?[133]
The uniqueness conjecture for Markov numbers[134]
Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[135]
Additive number theory
See also: Problems involving arithmetic progressions
Beal's conjecture
Fermat–Catalan conjecture
Goldbach's conjecture
The values of g(k) and G(k) in Waring's problem
Lander, Parkin, and Selfridge conjecture
Gilbreath's conjecture
Erdős conjecture on arithmetic progressions
Erdős–Turán conjecture on additive bases
Pollock octahedral numbers conjecture
Skolem problem
Determine growth rate of rk(N) (see Szemerédi's theorem)
Minimum overlap problem
Do the Ulam numbers have a positive density?
Algebraic number theory
Are there infinitely many real quadratic number fields with unique factorization (Class number problem)?
Characterize all algebraic number fields that have some power basis.
Stark conjectures (including Brumer–Stark conjecture)
Kummer–Vandiver conjecture
Greenberg's conjectures
Computational number theory
Integer factorization: Can integer factorization be done in polynomial time?
Prime numbers
vte
Prime number conjectures
Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.
Brocard's Conjecture
Catalan's Mersenne conjecture
Agoh–Giuga conjecture
Dubner's conjecture
The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
New Mersenne conjecture
Erdős–Mollin–Walsh conjecture
Are there infinitely many prime quadruplets?
Are there infinitely many cousin primes?
Are there infinitely many sexy primes?
Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
Are there infinitely many Wagstaff primes?
Are there infinitely many Sophie Germain primes?
Are there infinitely many Pierpont primes?
Are there infinitely many regular primes, and if so is their relative density e − 1 / 2 {\displaystyle e^{-1/2}} e^{-1/2}?
For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?
Are there infinitely many Cullen primes?
Are there infinitely many Woodall primes?
Are there infinitely many Carol primes?
Are there infinitely many Kynea primes?
Are there infinitely many palindromic primes to every base?
Are there infinitely many Fibonacci primes?
Are there infinitely many Lucas primes?
Are there infinitely many Pell primes?
Are there infinitely many Newman–Shanks–Williams primes?
Are all Mersenne numbers of prime index square-free?
Are there infinitely many Wieferich primes?
Are there any Wieferich primes in base 47?
Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?
For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[136]
Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[137]
Are there infinitely many Wilson primes?
Are there infinitely many Wolstenholme primes?
Are there any Wall–Sun–Sun primes?
For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
Is every Fermat number 22n + 1 composite for n > 4 {\displaystyle n>4} n>4?
Are all Fermat numbers square-free?
For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
Artin's conjecture on primitive roots
Is 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?
Is 509,203 the lowest Riesel number?
Fortune's conjecture (that no Fortunate number is composite)
Landau's problems
Feit–Thompson conjecture
Does every prime number appear in the Euclid–Mullin sequence?
Does the converse of Wolstenholme's theorem hold for all natural numbers?
Elliott–Halberstam conjecture
Problems associated to Linnik's theorem
Find the smallest Skewes' number
Set theory
The problem of finding the ultimate core model, one that contains all large cardinals.
If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, was obtained by Shelah using his pcf theory.
Woodin's Ω-hypothesis.
Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
(Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
Does there exist a Jónsson algebra on ℵω?
Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
Does the Generalized Continuum Hypothesis entail ♢ ( E cf ( λ ) λ + ) {\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})} {\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})} for every singular cardinal λ {\displaystyle \lambda } \lambda ?
Does the Generalized Continuum Hypothesis imply the existence of an ℵ2-Suslin tree?
Is OCA (Open coloring axiom) consistent with 2 ℵ 0 > ℵ 2 {\displaystyle 2^{\aleph _{0}}>\aleph _{2}} {\displaystyle 2^{\aleph _{0}}>\aleph _{2}}?
Topology
The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.
Baum–Connes conjecture
Borel conjecture
Hilbert–Smith conjecture
Mazur's conjectures[138]
Novikov conjecture
Telescope conjectures
Unknotting problem
Volume conjecture
Whitehead conjecture
Zeeman conjecture
Problems solved since 1995
Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.
Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)[139] [140]
Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)[141][142]
Connes embedding problem (Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, 2020)
Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)[143]
Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)[144]
McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel)(Karim Adiprasito, 2018)[145][146]
Yau's conjecture (Antoine Song, 2018)[147]
Pentagonal tiling (Michaël Rao, 2017)[148]
Burr–Erdős conjecture (Choongbum Lee, 2017)[149]
Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)[150][151]
Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (Alireza Abdollahi, Maysam Zallaghi, 2015)[152]
Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)[153]
Erdős discrepancy problem (Terence Tao, 2015)[154]
Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)[155]
Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
Anderson conjecture (Cheeger, Naber, 2014)[156]
Gaussian correlation inequality (Thomas Royen, 2014)[157]
Goldbach's weak conjecture (Harald Helfgott, 2013)[158][159][160]
Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)[161][162] (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic K S r {\displaystyle KS_{r}} {\displaystyle KS_{r}} and K S r ′ {\displaystyle KS'_{r}} {\displaystyle KS'_{r}} conjectures, Bourgain-Tzafriri conjecture and R ϵ {\displaystyle R_{\epsilon }} {\displaystyle R_{\epsilon }}-conjecture)
Virtual Haken conjecture (Agol, Groves, Manning, 2012)[163] (and by work of Wise also virtually fibered conjecture)
Hsiang–Lawson's conjecture (Brendle, 2012)[164]
Willmore conjecture (Fernando Codá Marques and André Neves, 2012)[165]
Beck's 3-permutations conjecture (Newman, Nikolov, 2011)[166]
Ehrenpreis conjecture (Kahn, Markovic, 2011)[167]
Hanna Neumann conjecture (Mineyev, 2011)[168]
Bloch–Kato conjecture (Voevodsky, 2011)[169] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture[170][171][172])
Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)[173]
Density theorem (Namazi, Souto, 2010)[174]
Hirsch conjecture (Francisco Santos Leal, 2010)[175][176]
Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)[177]
Atiyah conjecture (Austin, 2009)[178]
Kauffman–Harary conjecture (Matmann, Solis, 2009)[179]
Surface subgroup conjecture (Kahn, Markovic, 2009)[180]
Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[181]
Cobordism hypothesis (Jacob Lurie, 2008)[182]
Full classification of finite simple groups (Harada, Solomon, 2008)
Geometrization conjecture, proven by Grigori Perelman[183] in a series of preprints in 2002–2003.[184]
Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)[185][186][187]
Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[188]
Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)[189]
Erdős–Menger conjecture (Aharoni, Berger 2007)[190]
Road coloring conjecture (Avraham Trahtman, 2007)[191]
Spherical space form conjecture (Grigori Perelman, 2006)
The angel problem (Various independent proofs, 2006)[192][193][194][195]
Nirenberg–Treves conjecture (Nils Dencker, 2005)[196][197]
Lax conjecture (Lewis, Parrilo, Ramana, 2005)[198]
The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)[199]
Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)[200]
Robertson–Seymour theorem (Robertson, Seymour, 2004)[201]
Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)[202] (and also Alon–Friedgut conjecture)
Green–Tao theorem (Ben J. Green and Terence Tao, 2004)[203]
Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)[204]
Carpenter's rule problem (Connelly, Demaine, Rote, 2003)[205]
Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)[206][207]
Milnor conjecture (Vladimir Voevodsky, 2003)[208]
Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)[209]
Nagata's conjecture (Shestakov, Umirbaev, 2003)[210]
Kirillov's conjecture (Baruch, 2003)[211]
Poincaré conjecture (Grigori Perelman, 2002)[183]
Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)[212]
Kouchnirenko’s conjecture (Haas, 2002)[213]
Vaught conjecture (Knight, 2002)[214]
Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)[215]
Catalan's conjecture (Preda Mihăilescu, 2002)[216]
n! conjecture (Haiman, 2001)[217] (and also Macdonald positivity conjecture)
Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)[218]
Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)[219]
Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)[220]
Erdős–Stewart conjecture (Florian Luca, 2001)[221]
Berry–Robbins problem (Atiyah, 2000)[222]
Erdős–Graham problem (Croot, 2000)[223]
Honeycomb conjecture (Thomas Hales, 1999)[224]
Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)[225]
Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)[226][227]
Lafforgue's theorem (Laurent Lafforgue, 1998)[228]
Kepler conjecture (Ferguson, Hales, 1998)[229]
Dodecahedral conjecture (Hales, McLaughlin, 1998)[230]
Ganea conjecture (Iwase, 1997)[231]
Torsion conjecture (Merel, 1996)[232]
Harary's conjecture (Chen, 1996)[233]
Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)[234][235]
See also
List of conjectures
List of unsolved problems in statistics
List of unsolved problems in computer science
List of unsolved problems in physics
Lists of unsolved problems
Open Problems in Mathematics
The Great Mathematical Problems
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Bruhn, Henning; Schaudt, Oliver (1998). "A proof of the dodecahedral conjecture". arXiv:math/9811079.
Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]". Inventiones Mathematicae. 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.
Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.
Further reading
Books discussing problems solved since 1995
Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 978-1-84115-791-7.
O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 978-0-471-08601-7.
Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 978-0-19-280722-9.
Books discussing unsolved problems
Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 978-1-56881-111-6.
Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 978-0-387-97506-1.
Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 978-0-387-20860-2.
Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 978-0-88385-315-3.
du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 978-0-06-093558-0.
Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 978-0-309-08549-6.
Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 978-0-691-11748-5.
Ji, Lizhen; Poon, Yat-Sun; Yau, Shing-Tung (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 978-1-57146-278-7.
Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. ISSN 1609-3321. S2CID 11845578. Zbl 1066.11030.
Mazurov, V. D.; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". arXiv:1401.0300v6 [math.GR].
The Sverdlovsk Notebook is a collection of unsolved problems in semigroup theory.[1][2]
Formulation of 50 {\displaystyle 50} 50 unresloved problems for infinite Abelian groups are depicted in the book[3]
The list of 17 {\displaystyle 17} 17 unresolved problems for Combinatorial Geometry are depicted in the book[4].
Several dozens of unresolved problems for Combinatorial Geometry are depicted in the book[5].
Many unresolved problems for Graph theory are depicted in the article[6].
The list of several unresolved problems converning Maler Conjecture are depicted in the book [7].
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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